# Overview: Modeling Forces

#### Prerequisites

In our section on Newton's Laws we established that a good way to treat motion is to think in terms of objects and interactions (see Object egotism and System Schema Introduction). When looking at individual objects, the interaction play out as forces that the interacting objects exert on each other. What we now need to do is examine many interacting systems and build quantitative models of force. (See Quantifying impulse and force,) Now what we need to do is decide what kinds of forces an object might feel.

Recall that the reason we want to do this is to predict the future — to understand how things move — and what's going on when they don't move.  If we know an object's starting position and velocity, Newton's second law will tell us how that velocity will change if we know all the forces acting on the object. Since Newton's second law can be written as a differential equation, we can use it as a stepping rule to predict where the object will be in the future. Knowing what controls an object's motion can be quite useful, whether you are trying to throw a large rock into a castle, to put a rocket in orbit around Saturn, to understand how a fish has to evolve to slurp up food, or why certain ions pass through a membrane and others don't.

For working with macroscopic objects, we want to build on our experience. We know of two very different ways that an object's velocities can be changed: The first is to touch it (hit it with a hammer), and the second is to let some action-at-a-distance force act on it (drop it). This suggests that we consider two broad classes of macroscopic forces: touching and non-touching.

## Touching forces

To think about how objects interact to change each other's velocities, we have to keep in mind object egotism — that we have decided that objects are in general dumb as posts (indeed, they may be posts) and only can respond to things that act on them directly at the instant that interaction takes place. So: thinking like an object, what are the different ways that an object can interact with you by touching?  The key — to the object — is in what direction does the force act?  We'll define three kinds of touching forces: normal, tension, and friction.

1. Normal forces — The word "normal" is an old word for "perpendicular". We call a touching force on an object "normal" if it is perpendicular to the surface of the object and pushing in. Think of pushing on the object with your hand — or a hammer.
2. Tension forces — We call a force a tension force if it pulls out on the surface of the object. Think of somebody gluing a string to the object and pulling on the string.
3. Resistive forces — When two objects rub surfaces, they exert forces on each other that resist the sliding. The result is a force on each object that is parallel to the surface of interaction and that opposes the sliding.  (Note: This does NOT mean that it "opposes the motion".) When the two surfaces are solids, the force is called friction.  When one (or both) of the surfaces are fluids, the force is called viscosity.  If a solid moves through a gas, it also experiences drag. Friction is independent of the relative velocity of the two surfaces, viscosity is proportional to the relative velocity of the two surfaces, and drag is proportional to the relative velocity squared.

There are subtle issues about each of these forces. We consider each of these in more detail on separate pages (available from the links connected to the force name above).

## Non-touching forces

You've had experience with the fact that things can be made to move without anything touching it directly since you were an infant. Everywhere on the earth, every object feels a downward pull that we give the name "weight". (We're going to save the name "gravity" for something else.) When you take a sock out of the dryer (if you haven't used dryer sheets) it might jump out of the basket and stick to your shirt. You probably have seen magnets pushing other magnets around at a distance from the time you were in kindergarten or before. These three forces are more fundamental than touching forces. At the microscopic level, touching forces can be seen to be the result of electric forces (like the sock in the dryer) combined with quantum mechanics. The non-touching forces we'll be concerned with here are

1. Weight — An object's weight is the result of the gravitational force that acts between the object and the earth. Gravitational forces act between all objects, but gravity is such a weak force that it's only really relevant when planet-sized bodies are around. Since we only have one, we only have to worry about the earth. (We might consider some properties of gravity and the implications for planets, moons, solar systems, and galaxies later down the line.)
2. Electric forces — As you know, all objects are made of atoms and all atoms contain two kinds of charged particles — electrons and protons. Charged particles exert electric forces on each other at a distance. Since the kinds of charges are nearly perfectly balanced in everyday matter, these forces were not thought to be important for thousands of years. But they underlie all of the fundamental biological processes that make life possible.
3. Magnetic forces — The basic charged particles also exert another kind of long-range touching force on each other — magnetic forces. Every electron and proton is not only a separate electric charge, it is a little bar magnet.  But the magnetic force tends to be significantly weaker than the electric force in most biological cases. It is, however, of great importance in engineering — where it makes possible the entire structure of the generation and transmission of electrical energy that has an immense impact on our everyday lives. Some chemical and biological measurement tools (such as NMR, fMRI, and magnetic spectrometers) depend strongly on magnetic forces.

## Labeling forces

In many examples in this class we will have LOTS of forces. It's therefore very important to have a way to label each force carefully so that we know what we are talking about. We'll use three conventions.

1. We will call every force "$F$" to remind ourselves that the symbol represents a force;
2. We will put a subscript on the $F$ to tell us what object is causing the force and what object is feeling it; and
3. We will put a superscript from the following set of labels: (N, T, f, W, E, M) to tell us whether the force is (Normal, Tension, friction, Weight, Electric, Magnetic).

Thus, if block A is pushing up against block B, we will write the force felt by block B as $F^N_{A \rightarrow B}$.

Joe Redish 9/18/11

Article 524